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The radius of the suns orbit around the barycenter of the solar system is often measured to be roughly the radius of the sun. Based on the same type of bias that has made humans miss things about their solar system because they are looking at it from the vantage point of Earth orbiting inside the system, is it possible that the suns orbit around the barycenter is actually much larger?

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    $\begingroup$ "bias that has made humans miss things about their solar system" What kind of things do you have in mind. What things have we missed? $\endgroup$ – James K Dec 14 '19 at 8:24
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    $\begingroup$ You see bias because of vantage point all the time in all sciences. Geocentrism is one example. I think it was Marshal McLuhan who said "the fish does not know it's swimming in the ocean". If ancient cultures believed the Earth was flat (as animals have evolved mainly to navigate a flat surface, "place cells" adapted for that), another good example. In my opinion very characteristic of human condition, though maybe others see it differently. $\endgroup$ – Kornelia Dec 14 '19 at 8:31
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    $\begingroup$ Those are both very old and pre-science beliefs. Do you have anything more recent? Perhaps in the last 50 or 100 years. $\endgroup$ – James K Dec 14 '19 at 8:41
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    $\begingroup$ James K - A lot of things take a years to discover if we're wrong about them. Lobotomy won the Nobel Prize in 1949, and was common well-into the 1970s, unfortunately the peer review only served to make more people lose brain functionality. Lysenkoism was state-sponsored into the 1960s. And we've got Cutter Laboratories to thank for anti-vax movement. Some of our current ideas will probably be wrong or missing key info, it would be nice to be 100% right about the world, but the next Thomas Edison will probably have a "Don't talk to me about X-rays" moment too. $\endgroup$ – IKM Dec 15 '19 at 5:51
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    $\begingroup$ Unless you mean something other than "could there be missing, asymmetrically distributed mass in the solar system", then you should reword your question. There are lots of Q+A on the site about possible unseen planets. $\endgroup$ – Rob Jeffries Dec 15 '19 at 9:27
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This is basically equivalent to asking whether there's an undetected massive object in the outer Solar System.

Starting from the basics, the location of the centre-of-mass of a system of point masses is given by

$$\mathbf{X} = \frac{\sum_{i} m_{i} \mathbf{x}_{i}}{\sum_{i}m_{i}}$$

Note that the quantities in bold are vector quantities. Without loss of generality, we can use a reference frame where $\mathbf{X} = \mathbf{0}$, i.e. the origin of the coordinate system is the centre-of-mass. For a two-body system, this gives:

$$m_{1} \mathbf{x}_{1} = -m_{2} \mathbf{x}_{2}$$

So the second particle is located in the opposite direction of the first from the centre-of-mass, with the ratio of the distances $r_{1} = \left\| \mathbf{x}_{1} \right\|$, $r_{2} = \left\| \mathbf{x}_{2} \right\|$ depending on the mass ratio $q = m_{2} / m_{1}$:

$$r_{1} = qr_{2} = \left( \frac{q}{1+q} \right) r$$

Where $r = r_{1} + r_{2}$ is the total separation between the two objects. This means that the trajectories of the two objects around the centre-of-mass are scaled mirror-images of each other. So the semimajor axis of the reflex orbit of the Sun $a_{1}$ for a planet of mass $m_{2}$ with orbital semimajor axis $a$ is given by:

$$a_{1} = qa_{2} = \left( \frac{q}{1+q} \right) a$$

This orbit is larger for more massive planets (higher $q$) and for wider planetary orbits (higher $a$). Working this out for the major planets, the largest reflex orbit is due to Jupiter (1.06 solar radii), followed by Saturn (0.59 solar radii), Neptune (0.33 solar radii) and Uranus (0.18 solar radii).

If Planet Nine exists with the predicted properties given by Batygin et al. (2019), a ~5–10 Earth mass planet in a ~400–800 AU orbit would give a reflex orbit of ~1 to ~5 solar radii, which would likely be the largest contribution to the Sun's orbit around the Solar System barycentre.

The hypothesis that the Sun may have a distant binary companion or a giant planet in the Oort cloud (which would both be more massive than the proposed Planet Nine, and much further out), which has variously been termed "Nemesis" or "Tyche" would result in an even larger orbit around the Solar System barycentre. So far, these hypotheses have not held up to observational evidence.

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  • $\begingroup$ I was more thinking based on observational evidence. For example, assuming there actually was an undetected object (or some other explanation), would the observational proof of that have been missed? $\endgroup$ – Kornelia Dec 14 '19 at 11:34
  • $\begingroup$ Barycenter seems quite difficult to observe since all surrounding objects (all stars, galaxies, everything) move along with it, from reference frame of Earth. $\endgroup$ – Kornelia Dec 14 '19 at 11:36
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    $\begingroup$ Please read the links you were provided so that we all can step forward. The concept of the barycenter and its influence on coordinate systems is explained there. For example, in order to calculate a planet's position, e.g. for navigation, one does a simple vector addition: "observatory->(earth's center) + (earth's center)->(earth-moon barycenter) + (earth-moon barycenter)->(solar system barycenter) + (solar system barycenter)->planet. ". Source: cv.nrao.edu/~rfisher/Ephemerides/ephem_use.html#view. This works with pretty high precision since a long time. $\endgroup$ – user31179 Dec 14 '19 at 16:45
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Our understanding of orbits is not based simply on astronomical observation, it's based on physics as well.

In ancient times, we didn't have a very complete understanding of the physics of the universe. Philosophers (as what passed for scientists were called then) had many preconceived notions (e.g. all motion in the universe was either straight or circular), and they tried to fit their observations with them. It didn't feel like the Earth was moving, but they could see stars and planets moving, so they assumed the earth was stationary; to explain the complicated movements of planets they came up with things like epicycles.

Since then we figured out the heliocentric organization of the solar system (no more epicycles needed), then later that the solar system is part of a galaxy, which is part of a much larger universe, and scientists like Galileo, Newton, and Einstein refined our understanding motion and gravity. Astronomical observations on varying scales confirmed that these understandings were substantially correct (although when they got to galactic scales they discovered anomalies, leading to the notion of dark matter -- if this sounds like epicycles all over, read What Astronomers Wish Everyone Knew About Dark Matter And Dark Energy).

In addition to this, our observational technology has advanced. So even though we're observing from a single vantage point, we're able to detect motion in multiple directions. For instance, the Doppler effect can be used to detect motion to or away from the Earth (this is one of the techniques used to detect exoplanets, and it should be far more accurate for the Sun because it's so much closer). So we can map the precise motion of the Sun, and compare this with the expected motion around the barycenter with the known planets. If there were a significant difference, we'd go looking for an explanation; there might be another planet we haven't discovered (Neptune and Pluto were both discovered by looking for the sources of disturbances in other orbits), or some as-yet-unknown physical effect (e.g. General Relativity explained the anomalous orbit of Mercury).

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    $\begingroup$ A distant planet would not produce a measurable Doppler effect (or demonstrate otherwise). A search for "planet 9" continues. If it were at 1000au and 10 Earth masses, the barycentre would be 6 solar radii from the Sun. The Sun would orbit this barycentre every 32,000 years at 2.8 cm per second. $\endgroup$ – Rob Jeffries Dec 15 '19 at 9:20
  • $\begingroup$ The Doppler effect is more precise the faster an object is moving towards or away from us. For example, Barnard's Star has a high blueshift because it is currently getting closer to the solar system very quickly. Since the Earth's movement in the solar system is mostly perpendicular to the direction of the Sun, this doesn't help us much. $\endgroup$ – CJ Dennis Dec 16 '19 at 4:20
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The barycentre of the whole solar system is inside the Sun roughly half the time, and outside the other half.

From Wikipedia's article:

To calculate the actual motion of the Sun, only the motions of the four giant planets (Jupiter, Saturn, Uranus, Neptune) need to be considered. The contributions of all other planets, dwarf planets, etc. are negligible. If the four giant planets were on a straight line on the same side of the Sun, the combined center of mass would lie about 1.17 solar radii or just over 810,000 km above the Sun's surface.

The associated image shows that the barycentre is not stationary in regards to the Sun, but moves. More properly, the Sun moves around the barycentre. Sometimes the barycentre is almost at the centre of the Sun and sometimes it's about twice the radius of the Sun from its centre.

enter image description here

For the barycentre to be much further from the centre of the Sun than currently believed, there would have to be an undiscovered Uranus-to-Jupiter mass planet in the solar system close enough to the Sun to make a difference. This hypothetical body would have an effect on the rest of the solar system. This is how some of the outer planets were discovered: some orbits were being noticeably perturbed by something. The search for the something led to the discovery of another planet.

If it turns out that Planet Nine exists and is discovered and is a super-Earth with a mass around five to ten times that of the Earth (compared with 14.5 for Uranus and 17 for Neptune), it would be so far away that its effects on the barycentre would be negligible. The four giant planets are much closer, and much more massive, so have a much more noticeable effect.

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    $\begingroup$ Last paragraph is incorrect. $m_p a_p = M_{\odot} a_{\odot}$. The further away the planet is, the greater it's effect on the barycentre position. $\endgroup$ – Rob Jeffries Dec 15 '19 at 9:11
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    $\begingroup$ Gravity does not appear in the formula, only masses and distance. Like, for instance, these scales of old where one could shift the counterweight ... $\endgroup$ – user31179 Dec 15 '19 at 11:12
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    $\begingroup$ @RoryAlsop the formula I have given is how you calculate the centre of gravity. It also features in the correct, but still not accepted, answer by antispinwards. This answer is plain wrong and yet has attracted 4 upvotes. At least one since I made my comment. $\endgroup$ – Rob Jeffries Dec 15 '19 at 18:23
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    $\begingroup$ I see how the barycentre of a two body system increases by distance, but not for a system with more than two bodies. Otherwise we should consider all the dust, other stars, the black hole at the centre of our galaxy, other galaxies, etc. $\endgroup$ – CJ Dennis Dec 15 '19 at 22:39
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    $\begingroup$ For a system with multiple bodies, calculate where the centre of mass is and then add in the extra body and see how much it moves the barycentre. 1. We should consider dust in the solar system, but it is negligible. 2. Other stars and galaxies aren't part of the solar system, so don't affect our calculation of the solar system barycentre. $\endgroup$ – Rob Jeffries Dec 16 '19 at 7:32

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